Connectivity augmentation is fixed-parameter tractable

2026-05-12

AI Summary

This work addresses the problem of augmenting a graph by adding at most $k$ edges to achieve $\lambda$-vertex-connectivity or $\lambda$-edge-connectivity. The authors design fixed-parameter tractable (FPT) algorithms parameterized by $k$—and additionally by $\lambda$ in the vertex-connectivity case. They establish, for the first time, that the vertex-connectivity augmentation problem is FPT with respect to both $k$ and $\lambda$ for arbitrary $\lambda$, and extend the FPT result for edge-connectivity augmentation to the general setting without requiring additional assumptions. The proposed algorithms run in time $2^{O(k \log(k+\lambda))} n^{O(1)}$ for vertex-connectivity and $2^{O(k \log k)} n^{O(1)}$ for edge-connectivity, significantly broadening the scope of existing results.

Abstract

In the vertex connectivity augmentation problem, we are given an undirected $n$-vertex graph $G$, a set of links $L \subseteq \binom{V(G)}{2} \setminus E(G)$, and integers $λ$ and $k$. The task is to insert at most $k$ links from $L$ to $G$ to make $G$ $λ$-vertex-connected. We show that the problem is fixed-parameter tractable (FPT) when parameterized by $λ$ and $k$, by giving an algorithm with running time $2^{O(k \log (k + λ))} n^{O(1)}$. This improves upon a recent result of Carmesin and Ramanujan [SODA'26], who showed that the problem is FPT parameterized by $k$ but only when $λ\le 4$. We also consider the analogous edge connectivity augmentation problem, where the goal is to make $G$ $λ$-edge-connected. We show that the problem is FPT when parameterized by $k$ only, by giving an algorithm with running time $2^{O(k \log k)} n^{O(1)}$. Previously, such results were known only under additional assumptions on the edge connectivity of $G$.

Problem

• connectivity augmentation
• vertex connectivity
• edge connectivity
• fixed-parameter tractable
• graph augmentation

Innovation

• connectivity augmentation
• fixed-parameter tractable
• vertex connectivity
• edge connectivity
• parameterized algorithm